Related papers: Necessary and sufficient conditions for consistent…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
Fix $n\in\mathbb{N}$. Let $\mathbf{T}_n$ be the set of rooted trees $(T,o)$ whose vertices are labeled by elements of $\{1,...,n\}$. Let $\nu$ be a strongly connected multi-type Galton-Watson measure. We give necessary and sufficient…
This paper is a variation on the uniform spanning tree theme. We use random spanning forests to solve the following problem: for a Markov process on a finite set of size $n$, find a probability law on the subsets of any given size $m \leq…
Reconstructing the tree of life from molecular sequences is a fundamental problem in computational biology. Modern data sets often contain a large number of genes, which can complicate the reconstruction problem due to the fact that…
For a class of irreducible Markov chains with an infinitely countable set of states, we establish a new verifiable necessary and sufficient condition for recurrence and transience. We show that if one of the basic assumptions is not…
We provide a new approach for proving the indistinguishability of connected components of random one-or-two-ended oriented forests on unimodular random graphs. In particular, this approach leads to a new and simpler proof for the wired…
Given a gene tree and a species tree, ancestral configurations represent the combinatorially distinct sets of gene lineages that can reach a given node of the species tree. They have been introduced as a data structure for use in the…
Bayesian inference is now a leading technique for reconstructing phylogenetic trees from aligned sequence data. In this short note, we formally show that the maximum posterior tree topology provides a statistically consistent estimate of a…
A chief problem in phylogenetics and database theory is the computation of a maximum consistent tree from a set of rooted or unrooted trees. A standard input are triplets, rooted binary trees on three leaves, or quartets, unrooted binary…
Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $\beta>0$. We start at time 0 with the tree composed of a root…
We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order…
We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other…
The problem of structure estimation in graphical models with latent variables is considered. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider models where the…
We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the…
Species tree reconstruction from genomic data is increasingly performed using methods that account for sources of gene tree discordance such as incomplete lineage sorting. One popular method for reconstructing species trees from unrooted…
Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In [Daskalakis et al.'09], building on the…
We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on…
The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov…
We introduce and analyze a new class of monotone stochastic recursions in a regenerative environment which is essentially broader than that of Markov chains. We prove stability theorems and apply our results {to three canonical models in…
We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root,…